To determine the range of values for the third side of a triangle when the other two sides are given, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the sides of the triangle as follows:
- Side 1 = 7
- Side 2 = 11
- Side 3 = n (the unknown side)
According to the Triangle Inequality Theorem, the following three inequalities must be true:
- \( 7 + 11 > n \)
- \( 7 + n > 11 \)
- \( 11 + n > 7 \)
Now, we will solve these inequalities step by step:
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From the first inequality \( 7 + 11 > n \): \[ 18 > n \quad \text{or} \quad n < 18 \]
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From the second inequality \( 7 + n > 11 \): \[ n > 11 - 7 \quad \text{or} \quad n > 4 \]
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From the third inequality \( 11 + n > 7 \): \[ n > 7 - 11 \quad \text{or} \quad n > -4 \] (This condition is always satisfied, since \( n > 4 \) is more restrictive.)
Combining the results of the first two inequalities, we have: \[ 4 < n < 18 \]
So the range of values for the third side \( n \) is: \[ 4 < n < 18 \]
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